On Certain Identities in the Theory of Matrices

On Certain Identities in the Theory of MatricesAuthor(s): Henry TaberSource: American Journal of Mathematics, Vol. 13, No. 2 (Jan., 1891), pp. 159-172Published by: The Johns Hopkins University PressStable URL: http://www.jstor.org/stable/2369812 .

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On Certain Identities in the Theory of Matrices.

BY HENRY TABER, Clark University.

1. In this paper I consider some applications to the general theory of matrices of conceptions familiar in quaternions (the separation into scalar and vector parts, the conjugate, etc.) which I have extended in a previous paper in this Journal, Vol. XII, to matrices of order higher than the second. By this me'ans I find that the identical eq.uation, the identical relations between two mnatrices (constituting with the identical equation the catena of equations), and Sylvester's formula, may be exhibited as explicit identities, and that the law of latency is an immediate corollary of an explicitly identical proposition. Moreover, by this means I find expressions for the coefficients of the equations of the catena, as simple functions of the sum of the latent roots of the powers and products of powers of the matrices involved. I also prove the extension of the conceptions met with in quaternions without regarding the matrix as an operator linear in and distributive over the units of an algebra, as in establishing these conceptions in the paper above referred to: thus it will be shown that a matrix of the third or of higher order, like a matrix of order two (quaternion), is separable into a scalar and a non-scalar (or vector) part, and that the vector part of a matrix of order X is further separable into c - 1 sub-vector parts, which may be termed the first, second, etc., and (C - l)th vector parts. The separation of a matrix into a scalar and X vector parts is of much importance in the applications considered in this paper.* In regard to the extension of the conception of the conjugate,t it will be shown that a matrix of order X has ca - 1 conjugates which

* It will appear later that the identical equation is merely a corollary (immediate) of this separation of a matrix into its w parts.

t The term conjugate is employed in quaternions by Hamilton and Tait with two different significa- tions: 1, to denote a certain function of a quaternion; and 2, to denote a different function, the converse (Peirce) or transverse (Cayley and Sylvester) of a linear vector operator (matrix of the second or of the third order). (The converse of a given matrix is the matrix obtained by interchanging its rows and



160 TABER: On Certain Identities in the Theory of Matrices.

reduce, when ca = 2, to the single conjugate of quaternions; it will appear that the product of a matrix of order ca and its X - 1 conjugates is a scalar, equal to the content of the mnatrix; and, following the analogy of quaternions, I define the tensor of the matrix as the Pth root of this product, which is commutative. Further, I shall show that a matrix of order X may be represented as the product of its tensor into X - 1 versors, each versor being dependent upon a single parameter contained in all its X terms, or into a single versor dependent upon

- 1 parameters, each of which enters into its X terms. The functions of the parameters which appear in the versors of a matrix are siimple extensions of the trigonomnetrical functions, to which (or to the hyperbolic funictions) they reduce when a = 2.

I shall prove these propositions for matrices of the third order, but the method employed is perfectly general.

2. A matrix n of the third order, whose latent roots g1, g2., g3 are all distinct, may be represented as follows:

n- (g1 0 0 ) -, 0 g2 0

0 0 g3

where w is not vacuous.* Let

n ( all a12 a13 ), rii ( bl, b12 bl3 ), tzr1 ( Bl B21 B31 ), a21 a22 a23 b21 b22 b23 B12 B22 B32 31 a32 a33 b31 b32 b33 B13 B23 B33

columns.) A quaternion may be regarded as a matrix of the second order, and its conjugate (in Hamil- ton's and Tait's sense) as a matrix or linear vector operator, is not identical with its conjugate as a quaternion; so it is best to restrict the term in quaternions to its first signification. As a term is needed to express the same function of a matrix of order higher than the second, which the conjugate (with the first signification) is of a quaternion, or matrix of the second order, I employ the term conjugate for this purpose.

* From this form of n can obviously be obtained immediately demonstrations of the identical equation, of Sylvester's formula, and of the law of latency. It also gives an expression, more simple than Sylvester's formula, for any function of a matrix whose latent roots are all distinct, provided the function can be expressed in terms of positive integral powers of n, which is not always the case. For then we have

Fn=w(Fgq 0 0 )Wr- 0 Fg2 ?

0 o Fgt



TABER: On Certain Identities in the Theory of Matrices. 161

where B1l, B12, etc., denote, the first mninors of w with respect to bl, b12, etc., respectively, divided by the content of zi. If we put

{ (all- g) bil + al2b21 + a13b61 O0,

a2lb6l + (a22 - g1) b21 + a23b31 = 0 ,

a3lb1l + a32A21 + (a33- gl) b3l = 0,

(all -9g2) b12 + al2b2, + a=3b32 = 0,

a2lbl2 + (a22 - g2) b22 + a23b32 0,

a3lbl2 + a32b22 + (a33 - g2) b32 - 0,

((all - g3) bl3 + a12b23 + alb33 0,

6a21bl3 + (a22 -g3) b23 + a23633 = 0,

a3jb13 + a32b23 + (a33 - 93) b33 _0

the ratio of the constituents of each column of zi will be completely determined. From the group of three equations consisting of the first equation of each of these three sets, each transforined by putting the term containing g in the right-hand member, we obtain immnediately

all = gjb61B1j + g2b62B12 + g3b13B13,

a12 = gjbjB21+ g2b12B22 + g3b63B23,

a13 = gjb61B3j + g2bl2B32 + g3bl 3B33

three equations that, together with the six equations obtained in like manner from the second equations of each of the three sets, and from the third equations of each of the three sets, constitute the conditions necessary and sufficient that

n=zy(g1 0 O )W-1.*

0 g2 0 0 0 g3

If two of the latent roots are equal, as g= g2, then n may be represented, in the same way, provided the nullity of n - g, is two. In general, a matrix of order X is representable in like manner if its latent roots are all distinct, or if any latent root occurs m times, provided the nullity of the matrix, less that latent root, is m.t

*Compare this Journal, Vol. XII, p. 359. t This is readily seen by means of the above scheme of equations determining the constituents of ZY.

It may also be readily proved by regarding a matrix as an operator linear and distributive over the units of an algebra (i. e., as a linear unit, or vector operator). Thus, if of the three latent roots of

n, g1=Y2, then n% has three linearly independent axes (the only case in which the matrix is reprev sentable as above), only if n - gL annuls two linearly independent vectors; but thell I n - I (the dAtArminant of nf- a. ) has nullitv two. S this ounrrnal. Vol. XTT. n. 363.

21




3. The first conjugate of a matrix is obtained from the matrix, when in the form of (2), by a cyclic interchange of its latent roots; the second conjugate, by a repetition of this cyclic interchange, etc. Thus the first and second conjugates of the nonion n, denoted respectively by Iln and K2n, are

Kln =(g3 0 0 ) K K2n= u(g2 0 0 )u- o g10 0 g3 0 o o g2 0 0 g1

Obviously Kn = K1 (Kln), consequently we nmay dispense with the subscripts, and write K for K1, and K2 for K2. We have in the case of matrices of the third order, K. K2 = K2. K= K3 = 1, whereas in quaternions K2= 1.

It is very easy to see that the conjugate as defined here is identical, when = 2, with the quaternion conjugate. For by (2) any quaternion whose latent

roots g, and g2. are distinct, may be represented as

q w(g1 0 )ur',

jo g21

where w is now a matrix of the second order: RKq ( g2 0 )vx'vi( (g, 0 )+ 2(g1 +g2))zzr

0ogj o g21 = - q+ 2Sq= Sq- Vq,

which is the ordinary definition of the quaternion conjugate. Evidently, in general for inatrices of any order, as in quaternions,

K (Fn) = F(Kn). 4. If we put

e z1 0 ow1 o2t o o ) z-1 63 =- ty( o o)l oo o (o 1 0(0 0 0

o o0 0 o 0 0 0 1

then n = g91l + g2e2 + g3e3,

Kn =g31 + g9e2 + 263

K2n` g2= 1 + g3e2 + g1,l

and since the e's are idempotent aiid mutually nilfactorial, it follows that any symmetric function of the three con-jugate matrices n, Kn, K2n, (involving no




other matrix but unity) is the same symmetric function of their latent roots into El + E2 + F3 = 1 .

Fromn this theorem can immediately be derived the identical equation and Sylvester's formula as explicit identities, and the law of latency may be shown to be involved in an explicit identity:

A. For by the proposition just proved,

n3- (g1 + g2 + g3) n2 + (g2g3 + g3g1 + g,1g2) n-g1g2923

_n3-_(n -t Kn+K2n)n2 + ( In.K2n + K2n.n + n.Kn) n- n.Kn.12n -(n- n)(n - Rn)(n- K'n) = 0 .

B. By Sylvester's formula the expression for any function Fn is

Fgl. (g2 -g3) -Fg2 . (g1-g3) +EFg3 (g1 - 2) n (91

- 2) (91

- 3) (92 - 3)

Fgl . (g 2 g-) _ Fg2 . (g9 g- ) + Fq3 . (gl2 - 9y22) n (gl- g2)(g1 - g3)(92 - 93)

+ Fgl. (g22g3 - 929D -FPg. (g gl3 - g91g) + Fg3 * (9gg2 - g1g2) (g1 - g2)(g1 - g3)(92 - g3)

The coefficients of n2, n, and 1, are symmetric functions of the g's; hence, for the three latent roots, we may substitute n, Kn, and K"n, when the expression becomes linear in Fn, F(Kn), and F(K2n), and, on reducing, the coefficients of F(Kn) and F(K2n) will appear as zero, and the coefficients of Fn as

[Kn - K2n] n2 - [(Kn)2 - (K2n)2] n + [(K)2. K2n - Kn. (K2n)2] (n - Kn)(n - K2n)(Kn - K2n)

C. Since

(Fn)3_[Pgl+Fg2+Fg3] (Fn)2+[Fg2. Fg3+Fg3. Fg1+Fg1. Fg2](Fn)-Fgl. Fg2 . Fg3

- (Fn)3-[Fn + F (Kn) + F (K2n)] (Fn)2

+ [F (Kn). F(K2n) +F(K2n). Fn +Fn. F(Kn)](Fn) -Fn .F (Kn). F (K2n)

(Fn - Fn)(Fn - F (Kn))(Fn - F(K2n))= 0;

by the principle of this section the latent roots of Fn are found among the same functions of the latent roots of n, namely, Fgl, Fg2, Fg3. It might be, however, that these were not all latent roots of Fn; thus, Fg1 might occur twice, and Fg2 once as a latent root of Fn, and we might have

(Fn-Fgl)(Fn - Fg2) = 0.




In this case the latent roots of K(Fn) = F(Kn) and K2 (Fn) F(K2n) would also be Fgl, occurring twice, and Fg2: hence

Fn + F(Kn) + P(K2n) = 2Fg1 + Fg2.

But the left-hand member is a symmetric function of the latent roots of n equal to Fg1 + Fg2+ Fg3; hence Fgl - Fg3. In like manner, in any other case, we imay show that Fgl, Fg2, Fg3, are all latent roots of Fn.

5. The separation of a quaternion q with distinct latent roots into a scalar and a vector part may be arrived at by putting its latent roots equal, respectively, to a + b/- 1 and a - bV- 1, when

q '(a+bVI-1 0 ) a1=a+b.W( v-i 0 )'5l

0 a- b/-l 0 - 1 i

i.e., q.= a+bi,

where i is a norn-scalar square root of -1. It should be noted that T2q = a2+b is the content of q.

In the same way, if gl, g27 g3, are the three distinct latenit roots of n, and

g1= a+b +c,

g2= a +b +X2o,

g3= a + %2b + ac,

where X is an imaginary scalar cube root of unity, then

a.(1 0 O)z1+b.w(1 0 0)W-1+c.w (1 0 O) -1

0 1 0 0 x o 0 0 o

O 0 1 O O X2 0 0 X

a + bi + c2,

where i is here a non-scalar cube root of unity.* To select the scalar and non-

scalar parts of n we may employ the symbols S and V, as in quaternions;

* To complete the analogy between nonions and quaternions, the third power of the nonion units

should be scalar imaginary cube roots of unity, just as the quaternion units are fourth roots of unity whose squares are scalars (viz. - 1); and in the above separation of the nonion n, i should be a cube

root of X. A similar remark applies to the algebras derived from the vids of a matrix of order higher

than the third.




and to distinguish the first and second vector parts, we may employ the symbols VI and V27: thus

n Sn + Vn Sn + Vln + V2n.* We then have

Kn=Sn + X Vln + 27V2n, K2n Sn + a2 Vn + X V2n

By definition

T3n =n.Kn.KAn = n.K2n.Kn etc.t

a3 + b3 + c - 3abc = S3n + T3 Vln + T3 V2n - 3Sn. TV1n. TV2n.

Since, however, a3 + b3 + e - 3abc = gq12g3, hence Tn = 4' n where n denotes the determinant or content of n. Hence it follows that

T(nn') =Tn. Tn';

and then, if Un denote n + Tn,

U(nn')- Un. Un'.

6. As an immediate consequence of the separation of a quaternion into a scalar and vector, we have

(q - S0q)2 V2q

which is a form of the identical equation. In matrices of higher orders the identical equation may also be made to appear as an immediate corollary of the separation of. a mnatrix into its a parts. Thus in the case of a matrix of the third order,

(n - a)3 = (bi + oi2)3- = h + C3 + 3bc (bi + oi2);

i. e., (n -Sn)3=T3 Vln + T3V 2n + 3 TVTn. TV2n. (n -Sn).

* Neither V1 nor V2, as an operator, is distributive over a sum of nonions (as are obviously S and V). I take this occasion to correct an error made in my paper on the theory of matrices in this Journal, Vol. XII, p. 387, where formulae (based on the assumption that V1 and V2 are distributive) are given as the nonion analogues of the quaternion formula V (ap + pa) = 0 . The true nonion analogue of this formula involves only the symbols S and V. It is given in (8) of this paper.

t In quaternions K (qq') = Kq'. Kq. This is not a property of the symbol K for matrices of the third and higher orders. If, however, we define a new conjugate as follows:

m =-KRn. E2mK Km'1 T.mw-2 = Tm. Um-1,

which also reduces to the quaternion conjugate when w =2, then the property in question of the quaternion conjugate is true also of 3C; but neither this conjugate nor K is distributive over a sum of matrices, as is the quaternion conjugate over a sum of quaternions.




For a matrix of the fourth order, m a + bi + ci2 + di3, where i is now a non-scalar primitive fourth root of unity,

(m - a)4 = (bi + ci2 + di3)4 = C4 + d4 + 4bc2d -2b2d2

+ 2 (C2 + 2bd)(m a)2 +4c (b2 + d2)(m - a).

In the case of a matrix m of order 6), proceeding in a similar way, the ,th

power of the right-hand member (viz. Vm) will be the sum of a scalar (TW Vm) and scalar multiples of powers of Vm = m - Sm with exponents from one to f- 2.

7. On differentiating both members of the identity

(q- Sq)2= V2q,

we obtain the identical relation between q and any other quaternion dq = r, thus:

(q - Sq)(r - Sr) + (r- Sr)(q -Sq) = 2SVqIVr, i.e., qr + rq -2S.r 2Sr.q + 2SqKr = O.

It is readily seen that the scalar and vector parts of the left-hand member are separately zero.

Since the selective symbols V1 and V2 are not distributive, it is not possible to proceed in the same way to obtain the identical relation between two nonions from the identical equation in the form given in the last section; the same is true of matrices of higher orders. We may however dispense with the non- distributive selective symbols V1 and V2, etc., and employ only S and V, which are distributive for matrices of any order. Thus for the nonion n we have

VV13n= 3bc (bi + ci2) =- VS2n. Vn. /V3n= SV3n + VV3n = -SV3n + I S'V2n. Vn,

i e., (n - Sn)3 = SV3n + I SI2n.(n-Sn);

and reducing,

n3 -3Sn. n2 + 3 (S2n- 1, SV2n) n-(S3n + SV3n - Sn. SV2n) = O.

If we replace Vn throughout by n - Sn, we get the identical equation freed froin the symbol V, viz.

n3- 3Sn.n2 + - (3S2n -Sn2) n -( S3n- Sn2. Sn + Sn3) 0.




Likewise for the matrix m of the fourth order we have

VI4m = 5V4m + VV4m = SVm + _4 5V3nM. Vm + 28V2mn. V2m,

i. e., m4-4Snm.m3 + 6 (S2IM- 3 SV2M) m2 -4(S3rn - SV2m.Sm +-SV3m)m

+ (SIm -2SIm . S Vlm + 4 8m1. S:V3mn SV4m) = O . Here also we may substitute throughout in the coefficients m - Sm for Vm.

The same process may be extended to matrices of any order. Given in either form of this section, it is possible, by successive differentia-

tions of the identical equation of a matrix m of order ', to obtain the X - 1

identical relations between m and any other matrix m', which, together with the identical equations in m and m', constitute Sylvester's catena of equations. Thus in the case of a matrix of order three, differentiating the identical equation in n, we have, if dn = n',

(z2n' + nn'n + n'n2) -3Sn. (nn' + n'n) - 3 Sn'. n2

+ 3 (S2n- ISIV2n) n' + 3 (2SnSn' - SIVn Vn') n

- (3S2nSn' + 3S.V2nVn' - 35n.SVn I/n' - Sn'.SV2n) 0

Freed from the vector symnbol, this is

(n2n' + nn'n + ndn2) - 3Sn. (nn' + n'n) - 3Sn'. n2

+ 3 (3S2n - Sn2) n1 + 3 (3SnSn' - Snn') n

-(2 S2n . Sn'- - Sn2. Sn'- 9Sn. Snn + 35n2n') = 0.

Differentiating again with respect to n, regarding n' as a constant and putting the new dn = n', we obtain a new relation between n and n', which, on dividing through by the factor two, is what would be obtained by substituting in the above n for n' and n' for n. Differentiating once again with the same condition as before, we obtain the identical equation in n', on dividing through by the factor three.

8. In the last section it appeared that the identical equation of n could be obtained, by the proper substitution for Vn and the vector of powers of Vn, from the formula for VIPn, the nonion analogue of the quaternion formula VV2q = 0.t

* Since S and V are distributive, just as in quaternions, dSn =Sdn, dVn =Vdn. Moreover, as in quaternions, S (nn') =Sn. Sn'+ SVnVn', while V (nn') = Sn. VVn'+ Sn'. Vn + V. VnVn'; and SVnVn'= SVn'Vn, so that a cyclic interchange of a scalar product leaves it unaltered.

t See note on page 165 in reference to the nonion analogue of this formula.




On differentiating both members of the nonion analogue of this formnula, we obtain

V(P n. Vn' + Vn. Vn'. Vn + Vn'. V2n)-2 SV2n. Vn + 3SVn Vnl. Vn; and differentiating again, regarding n' as a constant,

V( Vn. V2n + Vn'. Vn. Vn' + V2n'. Vn) = 3S Vn Vn'. Vn + 3 SV2n'. Vn.

These formulae are the nonion analogues of the quaternion formula

V(VqVr + IrVq) = 0,

which can be obtained in like manner from VV2q = 0. Assuming these two formulae, the identical relations between n and n' may

be obtained as explicit identities by the proper substitutions in these formulae for Vn, Vn', and the vectors of products of their powers.

If, on the second differentiation of the formula for VV3n, we put dn - n", still regarding n' as a constant, we shall have

V( 1 Vn. Vn'. Vn") - 3 (S1/n Vn'Vn".Vn + S Vn" VTn. VW + S VnYu. Vn")

(where Z V1n. Vn'. Vn" denotes the sum of the products of Vn, Vn', Vn"l, in all possible orders), which is also an analogue of V( Vq Vr + Yr Vq) = 0. Making the proper substitutions in this formula, we obtain

nn'n" + nn"n' + n'nn" + n'n"n + n"nn' + n"n'n 38n. (n'n" + n"n') - 3Sn'. (n"n + nn") - 3Sn". (nn' + n'n)

+ 3 (3SnSn' - Snn') n" + 3 (3Sn"Sn - Sn"n) n' + 3 (3SnSn' - Snn') n'1 - (278n8n'Sn" - 9SnSn'n" - 9n'5n"n - 95n"Snn' + 3Snn'n" + 3Snn"n') = 0,

which miay, of course, also be obtained by differentiating twice the identical equation in n, regarding the first dn = n' as a constant, and putting the second dn = n".

9. The definition of Tn as the cube root of In is sufficieint to prove that a nonion is separable into the product of a tensor and a versor, since Inn' j = In . In' , and hence the product of two versors is a versor; but, to show the character of the versor part, we may proceed as follows: Since e0+,, eA0+A2n, eA2? + mnay-for proper values of 0 and n-have any ratios whatever, we may put

n=w(,gl 0 0)ur1=41gig2g3.zv( e0+ 0 0 )vf"1.

0 g2 0 0 e\0?+ \2 0

0 o 93 0 0 eA2O+? X|




Denoting 1 Eeo

+ [? + eko+A2- + eA20 + A] -I [e0+ n + %2eko+ k2 + 2ek2o+?Ax?

and 1- [ee+@ + ?+ek0+X2-q + %2eA20+,k'],

by fo (0, n), fi (0, n), and f (0, 7), respectively, then

n_=Tn.z( f (fl,Y)+fl(Y)+.f2 (0,7) 0 0 ) Z-1 o *fO (0, )+fl (0, 7?)+;2f2(0, ) 0

o 0 Oy,fo (0, 7)+X2f, (6, l)+f2 (6, 7?) - Tn (.fO (0, 0) +fl (0, 7) . i+.f2 (06 ,) . j2),

where i = Vin + TV1n is a non-scalar cube root of unity. If we denote fo (0, 0) =fo (0, 0), fi (0, 0)=2 (0, 0), f2 (0, ) f (0, 0) by

foo 0 f10 12 0, respectively, it is obvious that

n-=Tn (foO + f10 * i + 120 *i2)(fon + f2n * i + fin.i 2)

whence may be derived expressions for the functions fo(0, r), etc., in terms of oO I fon 7 etc., functions of a single parameter.

Since the latent roots of i and i2 are 1, 2, a2 henice, by Sylvester's forrmula, e"i- fOO + flO

. io + f2? .i2 el -Ol+gt J+fr

In ,27

n Tn.eOi?niO.

If, then,

n- Tn' (fo (0, , + 1i (0')i. + i+ (1(O', ). i2) = Tn' (10 +f10'.i +120 .i2)(Ion, +1f2nl.i +fin.i2),

.nn' = T(nn/)(fo (0 + 0',n +f)i (O + ?', n +f) i+12 (O + Of + f).i2) = T(nr')(1o (0 + 0') +1f (0 + 0'). i +f2 (0 + 0').i2)(fo (n + n'l)

+f2 (nl + rn ) . i +Ag (t + n 1) . J2) .

This formula gives the following formulae for the functions of the sum of two arguments:

1o(0 + 0', n + ') fo (0, n) fo (0', v') +f A(0, f)f2 (0' 7) +12(0, f)f (0', v'),

A (0 + 0', v + fo') 1 (0, v) -1 (0', v') + fl (0, f).o (0', v') + 2 (0, Ai).12 (0/', v), 2 (0 + 0', + '1fo (O? v) 12 (O', v') + 2( (?)1) *fo (v n') +1fl (O, fi).1l(, 0') ,

whence may be derived expressions for 1o (0 + 01), etc.

* In the paper referred to in the beginning of this article, in investigating the versor of a nonion, I have considered only fo0, fov, etc., functions of a single parameter; but, inadvertently, the above expression is put equal to Tn (f 0 (0 + rl) + fl (0+ r) . i +?f2 (0 +4 ) .2) which is false.

22




From the definition of these funictions it is evident that

fo ((O) =fo, fl (XO) = xfo? f2 (%O) = a2f20,

fo (XO, x2) = fo (O ),) f (2,0, x2) = Rf (O, ) f2 (A O x2n) = 2f2 (0, n)

.K..n = Tn (fo (%O , 2,20 + fi (%O, x2n) . +g A PL, 2u2) 2)

Tn (/o (XO) +fi (XO).i +f2 (X0) .i2)(fo (W2r) +12 (%2n) i +Af (2n) .i2)

Kn= Tnz(fo(X20 w7 ) +fi(20, 2rn) . +f2(2i, aA0 ).i2) =Tn (fo (X20) + f/ (2) * i +12 (t20) . i2)(fo (Xn) + 2 (%)*i + f/ (x%). il2)

This result may be obtained more simply from the equation n - Tn. ei+" i2 by means of the formiula (K (Fn) = F (Kn).

Since n-1 = (Kn .K2n) + T3n, hence

n-1 (Tn)-l (fO (-? 07- ) + fl ( - 0?, n)i +Af (- 0? t* n)

=(Tn)-1(fo(-0)+fi(-0) .i+f2A(-0)i2)(fo (-vn)+A2 (-v).i+fl(-_i).2)i)

10. In the Phil. Mag., Nov. 1883, Sylvester has given an application of his formula for any function of a matrix to the problemn to find an expression for the

-pth root of a quaternion; his results involve the trigonometric functions. A like process will give the pth root of a nonion in terms of the functions fo (0, eq), etc.; and a similar reinark applies to matrices of any order. But Sylvester's result may be obtained in a simplified formn and more easily as follows. If g, and g2 are the distinct latent roots of q, then for a proper value of 0 and of the quaternion zy, we may put

q = Tq.vi ( e 0 ) --1 =Tq.i ( e(O+4kI7r)V1 0 ) r- O e 1'-1 0 e-( - 027r)

1 1 (O4+ k, \-) 8 =P-TPq.i ( e p 0 ) zzr1

|~~ ~ ~~~ ?(

0

0.2.) -| 0 e ( ( p) 1 kO ( k+k-V (O 2k1

=Tllq.ep +-7r)+Sw(ept p 7r) - 0 ) .

0 e (*? klCffv

= p ( 0 + + ( ) (+ 0w)-Sin( + lr) .v -1

=pTq (cos (+ + 2k C)o+ sin ( + 27 )Uv TJq) = t 2k~ ( pp . ) 0 2 0 s .pTiiqyCos - ~7 +sin sUq) co +s'n-.UVq

p p 1 27r7t . 2 7op P




where p is a scalar pth root of unity. Whence the p2 pth roots of q are found amnong all possible combinations of any pth root of q with the p scalar pth roots of unity and the p quaternion pth roots of unity, whose vector part is parallel to Vq.

From this expression for qq we may readily derive 1

qp Aq + B,

where TIq sin (+ 27e7t) where ~~~~~A

= Tpq TP

- _ S )

A Tq sinO0

1 0 + 2k7_ .0 + 2Jcn cos 0 B pTPq cos p ) sin K sin )

which are much simpler expressions for A and B than those given by Sylvester in the paper above referred to.

Proceeding in the same way for the nonion n, we have

n Tn.zw (eO+ ?Ok1ffV7rzi/= o ?

o exo + A20 + 6Ok2ffV-1

o e020 + X + 6k 37/_ 1

1 1 6k1c7rl1"= 0 + O

.nP -Tipin. (e x 0 0 )-.r ( e P o0 0 ) a 6k2Sv-1 A0 + A2q

O e P 0 0 e P 0 __k ___V_ A20 + A*

O 0 e P0 O 0 e P

If we put ( 2k- k2 - ka)= = +Td,

(-k, 4 2k2 - k3') = M + %2k'l, (.- kk-2 + 2ksY= 2k + ?d,

6k1irV-1 * e P 0 0 ) m

6k2S8T O e P 0

6kg,rT/_ 1-

0 O0 e P

k,+ 42+ k2rVw-1 k+k 27rV1) =e r we P 0 0

AJC + 2KW'2 - 0 epkVk27r]v 0

0 e2kk 2w-



172 TABER: On Certain Identitie8 in the Theory of Matrices.

Hence, if we put 2k7t/-1 x and 2kY7?W-1 Yc,

nP=p TPn (fo( p+ ( p ) +f2 ( p p)

)f(p xp )O, x p '+pu)+ Q+O p+Oji2),

= p. TP8(o( +? + fi (X +,Xl+t) +f2 (X + . it+0 2)

where p is, as before, a scalar pth root of unity. Conrsequently the p3 pth roots of n may all be obtained by comnbining any pth root of n with the p scalar pth

roots of unity and the non-scalar pth roots of unity, whose first and second sub- vector parts are respectively scalar multiples of those of n. From this expres-

1

sion for np it is easy to derive

np=An'+ Bn+C,

but the coefficients A, B, C are not very simple expressions. WORCESTER, MASS., July 19, 1890.



On Certain Identities in the Theory of Matrices

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